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Transfinite diameter notions in ℂN and integrals of Vandermonde determinants

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Arkiv för Matematik

Abstract

We provide a general framework and indicate relations between the notions of transfinite diameter, homogeneous transfinite diameter, and weighted transfinite diameter for sets in ℂN. An ingredient is a formula of Rumely (A Robin formula for the Fekete–Leja transfinite diameter, Math. Ann. 337 (2007), 729–738) which relates the Robin function and the transfinite diameter of a compact set. We also prove limiting formulas for integrals of generalized Vandermonde determinants with varying weights for a general class of compact sets and measures in ℂN. Our results extend to certain weights and measures defined on cones in ℝN.

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References

  1. Berman, R. and Boucksom, S., Growth of balls of holomorphic sections and energy at equilibrium, Preprint.

  2. Berman, R. and Boucksom, S., Ball volume ratios, energy functionals and transfinite diameter for line bundles, Preprint.

  3. Berman, R. and Boucksom, S., Capacities and weighted volumes of line bundles, Preprint, 2008. arXiv:0803.1950

  4. Berman, R. and Boucksom, S., Equidistribution of Fekete points on complex manifolds, Preprint, 2008. arXiv:0807.0035

  5. Berman, R., Boucksom, S. and Nystrom, D. W., Convergence towards equilibrium on complex manifolds, Preprint.

  6. Berman, R. and Nystrom, D. W., Convergence of Bergman measures for high powers of a line bundle, Preprint, 2008. arXiv:0805.2846

  7. Bloom, T., Some applications of the Robin function to multivariable approximation theory, J. Approx. Theory92 (1998), 1–21.

    Article  MATH  MathSciNet  Google Scholar 

  8. Bloom, T., Weighted polynomials and weighted pluripotential theory, Trans. Amer. Math. Soc.361 (2009), 2163–2179.

    Article  MATH  MathSciNet  Google Scholar 

  9. Bloom, T., Bos, L., Christensen, C. and Levenberg, N., Polynomial interpolation of holomorphic functions in ℂ and ℂn, Rocky Mountain J. Math.22 (1992), 441–470.

    Article  MATH  MathSciNet  Google Scholar 

  10. Bloom, T. and Levenberg, N., Capacity convergence results and applications to a Bernstein–Markov inequality, Trans. Amer. Math. Soc.351 (1999), 4753–4767.

    Article  MATH  MathSciNet  Google Scholar 

  11. Bloom, T. and Levenberg, N., Weighted pluripotential theory in ℂN, Amer. J. Math.125 (2003), 57–103.

    Article  MATH  MathSciNet  Google Scholar 

  12. Bloom, T. and Levenberg, N., Asymptotics for Christoffel functions of planar measures, J. Anal. Math.106 (2008), 353–371.

    Article  MATH  MathSciNet  Google Scholar 

  13. Bloom, T., Levenberg, N. and Ma’u, S., Robin functions and extremal functions, Ann. Polon. Math.80 (2003), 55–84.

    Article  MATH  MathSciNet  Google Scholar 

  14. Deift, P., Orthogonal Polynomials and Random Matrices : A Riemann–Hilbert Approach, Amer. Math. Soc., Providence, RI, 1999.

    Google Scholar 

  15. Jedrzejowski, M., The homogeneous transfinite diameter of a compact subset of ℂN, Ann. Polon. Math.55 (1991), 191–205.

    MATH  MathSciNet  Google Scholar 

  16. Johansson, K., On fluctuations of eigenvalues of random Hermitian matrices, Duke Math. J.91 (1998), 151–204.

    Article  MATH  MathSciNet  Google Scholar 

  17. Klimek, M., Pluripotential Theory, Oxford University Press, New York, 1991.

    MATH  Google Scholar 

  18. Mehta, M. L., Random Matrices, 3rd ed., Pure Appl. Math. 142, Elsevier/Academic Press, Amsterdam, 2004.

    MATH  Google Scholar 

  19. Rumely, R., A Robin formula for the Fekete–Leja transfinite diameter, Math. Ann.337 (2007), 729–738.

    Article  MATH  MathSciNet  Google Scholar 

  20. Saff, E. and Totik, V., Logarithmic Potentials with External Fields, Springer, Berlin, 1997.

    MATH  Google Scholar 

  21. Siciak, J., Extremal plurisubharmonic functions in ℂn, Ann. Polon. Math.39 (1981), 175–211.

    MATH  MathSciNet  Google Scholar 

  22. Zaharjuta, V. P., Transfinite diameter, Chebyshev constants, and capacity for compacta in ℂn, Mat. Sb.96 (138) (1975), 374–389, 503 (Russian). English transl.: Math. USSR-Sb.25 (1975), 350–364.

    MathSciNet  Google Scholar 

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Authors and Affiliations

  1. University of Toronto, Toronto, Ontario, M5S 2E4, Canada

    Thomas Bloom

  2. Indiana University, Bloomington, IN, 47405, U.S.A.

    Norman Levenberg

Authors
  1. Thomas Bloom
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  2. Norman Levenberg
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Correspondence to Norman Levenberg.

Additional information

Bloom supported in part by an NSERC grant.

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Bloom, T., Levenberg, N. Transfinite diameter notions in ℂN and integrals of Vandermonde determinants. Ark Mat 48, 17–40 (2010). https://doi.org/10.1007/s11512-009-0101-9

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  • DOI: https://doi.org/10.1007/s11512-009-0101-9

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